Technically Exists

Free cancellation criterion

The free cancellation criterion requires that every portion of a vote must have an opposite which cancels it out while leaving the rest of the ballot free to express other preferences. Specifically, for every valid ballot and subset of the candidates there must exist at least one other ballot that represents the opposite set of preferences for those candidates, and adding these two ballots to an election must never change the result from one candidate in the subset to another. Furthermore, for every pair of candidates not in the subset, there must exist three such ballots with each one expressing a different preference between that pair of candidates.

More formally, a voting method mm passes the free cancellation criterion if for every ballot bb and subset of the candidates CC, all of the following hold:

  1. There exists a ballot bb' such that for each candidate cc in CC, b(c)=kb(c)b'(c) = k - b(c) for some integer constant kk. A ballot with this property is called an opposite to bb over CC.
  2. For every ballot bb' that is an opposite to bb over CC and every list of ballots b1,b2,,bnb_1, b_2, \dots, b_n, if m(b1,b2,,bn,b,b)m(b_1, b_2, \dots, b_n, b, b') is in CC, then either m(b1,b2,,bn)=m(b1,b2,,bn,b,b)m(b_1, b_2, \dots, b_n) = m(b_1, b_2, \dots, b_n, b, b') or m(b1,b2,,bn)m(b_1, b_2, \dots, b_n) is not in CC.
  3. For every unordered pair of candidates c1,c2c_1, c_2 not in CC, there must exist three ballots b1,b2,b3b'_1, b'_2, b'_3 that are opposites to bb over CC such that b1(c1)<b1(c2)b'_1(c_1) < b'_1(c_2), b2(c1)>b2(c2)b'_2(c_1) > b'_2(c_2), and b3(c1)=b3(c2)b'_3(c_1) = b'_3(c_2).

Note that this definition assumes that mm passes the anonymity criterion.

This criterion is intended to be a strengthened version of the partial cancellation criterion. While the partial cancellation criterion is passed by voting methods like negative voting and Borda count that aren’t fully expressive, the free cancellation criterion requires voting methods that pass it to allow the free expression of preferences among candidates not in the cancelling subset.

The free cancellation criterion implies the partial cancellation criterion, the opposite cancellation criterion, and the cancellation criterion.